Problem: Integrate. $\int\left(6e^x+\dfrac7x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $e^x+7\ln|x|+C$ (Choice B) B $e^x+7\ln(x)+C$ (Choice C) C $6e^x+7\ln|x|+C$ (Choice D) D $6e^x+7\ln(x)+C$
Answer: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(6e^x+\dfrac7x \right)dx \\\\ &=6\int e^x\,dx+7\int\dfrac1x \,dx \\\\ &=6e^x+7\ln|x|+C \end{aligned}$